Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 37-57. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2023-42-1-37-57
Research Article
Full text in Russian
MSC 26A33; 33E20

Contents of this issue

Read Russian Version

The Problem for a Mixed Equation with Fractional Power of the Bessel Operator

A. V. Dzarakhokhov ^{*, 1} , E. L. Shishkina ^{*, 2, 3}

¹Gorsky State Agrarian University, Russia, 37 Kirov St., Vladikavkaz 362040.
²Voronezh State University, Russia, 1 Universitetskaya Pl., Voronezh 394018.
³Belgorod State National Research University (BelGU), Russia, 85 Pobedy St., Belgorod 308015.

Abstract. Recently, of particular interest are partial differential equations containing a fractional order differential operator. Similar equations and problems for them find application in the theory of viscous elasticity, electrochemistry, control theory, modeling of epidemics and pandemics, and in various other areas. The present work is devoted to the solution of differential equations containing the Bessel operator of fractional degree. The article discusses the direct and inverse Meyer transforms, modified for the convenience of working with the Bessel operator of a fractional degree. For the considered Meyer transformation, a convolution is obtained. Using the Laplace and Poisson transformations, factorizations of the direct and inverse Meyer transformations are obtained. Using the considered modified Meyer transform, we find a solution to an ordinary differential equation with a Bessel operator of fractional degree. A nonlocal boundary value problem for a mixed parabolic-hyperbolic equation containing a fractional degree Bessel operator is considered. It is proved that, under certain conditions of smoothness of the input functions of the problem and the condition of conjugation on the dividing line of the regions of hyperbolicity and parabolicity, a regular solution of a nonlocal boundary value problem for a mixed parabolic-hyperbolic equation with a Bessel operator of fractional degree exists and is unique.

Key words: the Meyer transform, the Bessel operator of fractional degree, ordinary differential equations of fractional order, partial differential equations of fractional order.

Received: 14.03.2023; Revised: 20.03.2023; Accepted: 22.03.2023; First online: 15.04.2023

For citation. Dzarakhokhov A. V., Shishkina E. L. The problem for a mixed equation with fractional power of the Bessel
operator. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 37-57. EDN: DFSTCW. https://doi.org/10.26117/2079-6641-
2023-42-1-37-57.

Funding. Not applicable.

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing
the final version of the article in print. The final version of the manuscript was approved by all authors.

^*Correspondence: E-mail: azambat79@mail.ru, ilina_dico@mail.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Dzarakhokhov A. V., Shishkina E. L., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

References

  1. Bzhikhatlov H. G., Nakhushev A. M. On a boundary value problem for an equation of mixed parabolic-hyperbolic type, Dokl. AN SSSR, 1968, 183, 2, 261-264. (In Russian)
  2. Repin O. A., Kilbas A. A. An analog of the Bitsadze-Samarskii problem for a mixed-type equation with a fractional derivative, Dif. equations, 2003, 39, 5, 638-644. (In Russian)
  3. Voroshilov A.A., Kilbas A.A. The Cauchy problem for the diffusion-wave equation with the Caputo partial derivative, Differential Equations, 2006, 42, 5, 638-649.
  4. Khubiev K. U. Boundary value problem with shift for loaded hyperbolic-parabolic type equation involving fractional diffusion operator, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018, 28, 1, 82–90. (In Russian)
  5. Sprinkhuizen-Kuyper I. G. A fractional integral operator corresponding to negative powers of a certain second-order differential operator, J. Math. Analysis and Applications, 1979, 72, 674–702.
  6. McBride A. C. Fractional calculus and integral transforms of generalized functionsFractional calculus and integral transforms of generalized functions, London, Pitman, 1979, 179.
  7. Shishkina E. L., Sitnik S. M. On fractional powers of Bessel operators, Journal of Inequalities and Special Functions, Special issue to honor Prof. Ivan Dimovski’s contributions, 2017, 8, 1, 49–67.
  8. Kilbas A. A., Saigo M. H–Transforms. Theory and Applications, Florida: Chapman and Hall, Boca Raton, 2004, 408.
  9. Glaeske H. J., Prudnikov A. P., Skornik K. A. Operational calculus and related topics, Florida: Chapman and Hall, Boca Raton, 2006, 424.
  10. Prudnikov A. P., Brychkov Yu. A., Marichev O. I. Integrals and Series, Vol. 2, Special Functions, New York, Gordon & Breach Sci. Publ., 1992, 808.
  11. Shishkina E. L., Sitnik S. M. A fractional equation with left-sided fractional Bessel derivatives of Gerasimov-Caputo type, Mathematics, 2019, 7, 12, 21.
  12. Shishkina E. L., Sitnik S. M. Transmutations, singular and fractional differential equations with applications to mathematical physics, Cambridge, Academic Press, 2020, 592.
  13. Watson G. N. A Treatise on the Theory of Bessel Functions, Cambridge, University Press, 1922, 804.
  14. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and applications of fractional differential equations, Amsterdam, Elsevier, 2006, 523.

Information about authors


Dzarakhokhov Azamat Valerianovich – Senior Lecturer Dep. of Natural Sciences, Gorsky State Agrarian University, Vladikavkaz, Russia https://orcid.org/0000-0003-2231-4345.


Shishkina Elina Leonidovna – Dr. Sci. (Math. & Phys.) Associate Professor, Professor of the Dep. of Mathematical and Applied Analysis, Voronezh State University; Professor of the Dep. of Applied Mathematics and Computer Modeling, Belgorod State National Research University, Russia, https://orcid.org/0000-0003-4083-1207.